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Journal of Mathematical Chemistry

, Volume 49, Issue 8, pp 1587–1598 | Cite as

New Nordhaus-Gaddum-type results for the Kirchhoff index

  • Yujun Yang
  • Heping Zhang
  • Douglas J. Klein
Original Paper

Abstract

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index is the sum of resistance distances between all pairs of vertices in G. Zhou and Trinajstić (Chem Phys Lett 455(1–3):120–123, 2008) obtained a Nordhaus-Gaddum-type result for the Kirchhoff index by obtaining lower and upper bounds for the sum of the Kirchhoff index of a graph and its complement. In this paper, by making use of the Cauchy-Schwarz inequality, spectral graph theory and Foster’s formula, we give better lower and upper bounds. In particular, the lower bound turns out to be tight. Furthermore, we establish lower and upper bounds on the product of the Kirchhoff index of a graph and its complement.

Keywords

Resistance distance Kirchhoff index Nordhaus-Gaddum-type result 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceYantai UniversityYantaiPeople’s Republic of China
  2. 2.Mathematical Chemistry GroupTexas A&M University at GalvestonGalvestonUSA
  3. 3.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

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